In the paper, the （k-1）-traceable-nice （（k-1）-T-nice） and k-homogeneously-traceable-nice （k-HT-nice） sequence are defined similarly to the definition of k-Hamilton-nice （k-H-nice） and （k+1）-Hamilton-connec...In the paper, the （k-1）-traceable-nice （（k-1）-T-nice） and k-homogeneously-traceable-nice （k-HT-nice） sequence are defined similarly to the definition of k-Hamilton-nice （k-H-nice） and （k+1）-Hamilton-connected-nice （（k+1）-HC-nice） sequence. Therelationships among these four nice sequences are discussed. The main results are asfollows: Let<sub>η</sub>=(a<sub>1</sub>, a<sub>2</sub>,…, a<sub>k+1</sub> be a non-negative rational sequence, k≥2. （1） If η is（k+1）-HC-nice and a<sub>k+1</sub>=2, then η is k-HT-nice, （2） If η is k-HT-nice and a<sub>k+1</sub>=2,then η is （k-1）-T-nice, （3） If η is k-H-nice, then η is k-HT-nice. Meanwhile, four unsolvedproblems on these topics are proposed.展开更多
基金This project is supported by the National Natural Science Foundation of China.
文摘In the paper, the （k-1）-traceable-nice （（k-1）-T-nice） and k-homogeneously-traceable-nice （k-HT-nice） sequence are defined similarly to the definition of k-Hamilton-nice （k-H-nice） and （k+1）-Hamilton-connected-nice （（k+1）-HC-nice） sequence. Therelationships among these four nice sequences are discussed. The main results are asfollows: Let<sub>η</sub>=(a<sub>1</sub>, a<sub>2</sub>,…, a<sub>k+1</sub> be a non-negative rational sequence, k≥2. （1） If η is（k+1）-HC-nice and a<sub>k+1</sub>=2, then η is k-HT-nice, （2） If η is k-HT-nice and a<sub>k+1</sub>=2,then η is （k-1）-T-nice, （3） If η is k-H-nice, then η is k-HT-nice. Meanwhile, four unsolvedproblems on these topics are proposed.
